#6585795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39381	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6,9,10)(7,11,8,12)(13,14)(15,22)(16,20)(17,21)(18,19)(23,26)(24,28)(25,27);(1,2)(3,4)(6,10)(11,12)(14,19)(15,21)(16,24)(26,27);(2,4)(5,7,9,8)(6,11,10,12)(13,15,22,14,18,21,17,19)(16,25,26,28,24,23,27,20);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,18,24)(14,20,19,28)(15,23,21,25)(17,27,22,26);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,22)(14,21,19,15)(16,26,24,27)(20,23,28,25)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492991);